To these cases all combinations of concepts may be referred, and from them the entire doctrine of the judgment, its conversion, contraposition, equipollence, disjunction (this according to the third figure) may be deduced. From these also may be derived the properties of the judgment, upon which Kant based his pretended categories of the understanding, with the exception however of the hypothetical form, which is not a combination of concepts, but of judgments. A full account is given in the Appendix of “Modality,” and indeed of every property of judgments on which the categories are founded. With regard to the possible combinations of concepts which we have given, it has only further to be remarked that they may also be combined with each other in many ways. For example, the fourth figure with the second. Only if one sphere, which partly or wholly contains another, is itself contained in a third sphere, do these together exemplify the syllogism in the first figure, _i.e._, that combination of judgments, by means of which it is known that a concept which is partly or wholly contained in another concept, is also contained in a third concept, which again contains the first: and also, conversely, the negation; the pictorial representation of which can, of course, only be two connected spheres which do not lie within a third sphere. If many spheres are brought together in this way we get a long train of syllogisms. This schematism of concepts, which has already been fairly well explained in more than one textbook, may be used as the foundation of the doctrine of the judgment, and indeed of the whole syllogistic theory, and in this way the treatment of both becomes very easy and simple. Because, through it, all syllogistic rules may be seen in their origin, and may be deduced and explained. It is not necessary, however, to load the memory with these rules, as logic is never of practical use, but has only a theoretical interest for philosophy. For although it may be said that logic is related to rational thinking as thorough-bass is to music, or less exactly, as ethics is to virtue, or æsthetics to art; we must yet remember that no one ever became an artist by the study of æsthetics; that a noble character was never formed by the study of ethics; that long before Rameau, men composed correctly and beautifully, and that we do not need to know thorough-bass in order to detect discords: and just as little do we need to know logic in order to avoid being misled by fallacies. Yet it must be conceded that thorough-bass is of the greatest use in the practice of musical composition, although it may not be necessary for the understanding of it; and indeed æsthetics and even ethics, though in a much less degree, and for the most part negatively, may be of some use in practice, so that we cannot deny them all practical worth, but of logic even this much cannot be conceded. It is nothing more than the knowledge in the abstract of what